Answer:
The minimum diameter of the cylindrical tank needed to store the quantity needed to put out the fire is approximately 58.415 feet.
Step-by-step explanation:
A gallon equals 0.134 cubic feet. First, we determine the amount of water ([tex]Q[/tex]), measured in cubic feet, needed to put out the fire under the assumption that water is consumed at constant rate:
[tex]Q = \dot Q \cdot \Delta t[/tex] (1)
Where:
[tex]\dot Q[/tex] - Volume rate, measured in feet per minute.
[tex]\Delta t[/tex] - Time, measured in minutes.
If we know that [tex]\dot Q = 2000\,\frac{gal}{min}[/tex] and [tex]\Delta t = 120\,min[/tex], then the amount of water is:
[tex]Q = \left(2000\,\frac{gal}{min} \right)\cdot (120\,min) \cdot \left(0.134\,\frac{ft^{3}}{gal} \right)[/tex]
[tex]Q = 32160\,ft^{3}[/tex]
And the diameter of the cylindrical tank based on the capacity found above is determined by volume formula for a cylinder:
[tex]Q = \frac{\pi}{4}\cdot D^{2}\cdot h[/tex] (2)
Where:
[tex]D[/tex] - Diameter, measured in feet.
[tex]h[/tex] - Height, measured in feet.
If we know that [tex]Q = 32160\,ft^{3}[/tex] and [tex]h = 12\,ft[/tex], then the minimum diameter is:
[tex]D^{2} = \frac{4\cdot Q}{\pi\cdot h}[/tex]
[tex]D = 2\cdot \sqrt{\frac{Q}{\pi\cdot h} }[/tex]
[tex]D = 2\cdot \sqrt{\frac{32160\,ft^{3}}{\pi\cdot (12\,ft)} }[/tex]
[tex]D \approx 58.415\,ft[/tex]
The minimum diameter of the cylindrical tank needed to store the quantity needed to put out the fire is approximately 58.415 feet.
